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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sibfima | Structured version Visualization version GIF version |
Description: Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
Ref | Expression |
---|---|
sitgval.b | β’ π΅ = (Baseβπ) |
sitgval.j | β’ π½ = (TopOpenβπ) |
sitgval.s | β’ π = (sigaGenβπ½) |
sitgval.0 | β’ 0 = (0gβπ) |
sitgval.x | β’ Β· = ( Β·π βπ) |
sitgval.h | β’ π» = (βHomβ(Scalarβπ)) |
sitgval.1 | β’ (π β π β π) |
sitgval.2 | β’ (π β π β βͺ ran measures) |
sibfmbl.1 | β’ (π β πΉ β dom (πsitgπ)) |
Ref | Expression |
---|---|
sibfima | β’ ((π β§ π΄ β (ran πΉ β { 0 })) β (πβ(β‘πΉ β {π΄})) β (0[,)+β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sibfmbl.1 | . . . 4 β’ (π β πΉ β dom (πsitgπ)) | |
2 | sitgval.b | . . . . 5 β’ π΅ = (Baseβπ) | |
3 | sitgval.j | . . . . 5 β’ π½ = (TopOpenβπ) | |
4 | sitgval.s | . . . . 5 β’ π = (sigaGenβπ½) | |
5 | sitgval.0 | . . . . 5 β’ 0 = (0gβπ) | |
6 | sitgval.x | . . . . 5 β’ Β· = ( Β·π βπ) | |
7 | sitgval.h | . . . . 5 β’ π» = (βHomβ(Scalarβπ)) | |
8 | sitgval.1 | . . . . 5 β’ (π β π β π) | |
9 | sitgval.2 | . . . . 5 β’ (π β π β βͺ ran measures) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | issibf 33227 | . . . 4 β’ (π β (πΉ β dom (πsitgπ) β (πΉ β (dom πMblFnMπ) β§ ran πΉ β Fin β§ βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β)))) |
11 | 1, 10 | mpbid 231 | . . 3 β’ (π β (πΉ β (dom πMblFnMπ) β§ ran πΉ β Fin β§ βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β))) |
12 | 11 | simp3d 1144 | . 2 β’ (π β βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β)) |
13 | sneq 4633 | . . . . . 6 β’ (π₯ = π΄ β {π₯} = {π΄}) | |
14 | 13 | imaeq2d 6050 | . . . . 5 β’ (π₯ = π΄ β (β‘πΉ β {π₯}) = (β‘πΉ β {π΄})) |
15 | 14 | fveq2d 6883 | . . . 4 β’ (π₯ = π΄ β (πβ(β‘πΉ β {π₯})) = (πβ(β‘πΉ β {π΄}))) |
16 | 15 | eleq1d 2818 | . . 3 β’ (π₯ = π΄ β ((πβ(β‘πΉ β {π₯})) β (0[,)+β) β (πβ(β‘πΉ β {π΄})) β (0[,)+β))) |
17 | 16 | rspcv 3606 | . 2 β’ (π΄ β (ran πΉ β { 0 }) β (βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β) β (πβ(β‘πΉ β {π΄})) β (0[,)+β))) |
18 | 12, 17 | mpan9 507 | 1 β’ ((π β§ π΄ β (ran πΉ β { 0 })) β (πβ(β‘πΉ β {π΄})) β (0[,)+β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwral 3061 β cdif 3942 {csn 4623 βͺ cuni 4902 β‘ccnv 5669 dom cdm 5670 ran crn 5671 β cima 5673 βcfv 6533 (class class class)co 7394 Fincfn 8924 0cc0 11094 +βcpnf 11229 [,)cico 13310 Basecbs 17128 Scalarcsca 17184 Β·π cvsca 17185 TopOpenctopn 17351 0gc0g 17369 βHomcrrh 32868 sigaGencsigagen 33031 measurescmeas 33088 MblFnMcmbfm 33142 sitgcsitg 33223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7397 df-oprab 7398 df-mpo 7399 df-sitg 33224 |
This theorem is referenced by: sibfinima 33233 sitgfval 33235 sitgclg 33236 |
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